Limit theorem for large deviations probabilities of certain forms
type: Item
instance of: scholarly article
title: Limit theorem for large deviations probabilities of certain forms
zbMATH Open document ID: 0927.60043
DOI: 10.1007/BF02465581
published in: Lithuanian Mathematical Journal
publication date: 1999-06-14T00:00:00Z
review text: Consider the quadratic forms \(S_n=\sum^n_{k=1} \sum^{k-1}_{j=1} \beta_n^{k-j} \xi_k\xi_j\), \(n\geq 1\), where \(\{\beta_n\}\) is a real sequence such that \(\beta_n\to 1\) and \((\xi_k, {\mathcal F}_k, k\geq 1)\) is a uniformly bounded, square-integrable martingale difference sequence such that \(E(\xi^2_k \mid {\mathcal F}_{k-1}) =1\) a.s., \(k\geq 1\). Assume that \(n(1- \beta^2_n) \to\gamma \geq 0\). The main results present explicit real sequences \(\{x_n\}\) such that \[ \sup_{0\leq x\leq x_n} \left | {P(n^{-1} a_nS_n> x)\over 1-G (x)}-1 \right| \to 0 \text{ as } n\to \infty, \] where \(\{a_n\}\) is a real sequence and \(G\) is a distribution function (d.f.). This result holds, in particular, when \(\gamma= +\infty\) if \(a_n= (1-\beta^2_n)^{1/2}\) and \(G\) is the standard normal d.f. and, in general, with \(a_n=1\) and \(G(x)= P[\int^1_0 Y(t)dW(t)\leq x]\), where \(W(t)\) is a standard Wiener process and \(Y(t)\) is an Ornstein-Uhlenbeck process.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1296187
zbMATH Keywords: martingale
zbMATH Keywords: quadratic form
zbMATH Keywords: large deviations
zbMATH Keywords: Hilbert space
zbMATH Keywords: uniform convergence
zbMATH Keywords: weak convergence
author: Alfredas Račkauskas
reviewed by: R. James Tomkins
MaRDI profile type: Publication
cites work: On probabilities of large deviations in Banach spaces
cites work: Exact convergence rates in some martingale central limit theorems
cites work: Asymptotic inference for nearly nonstationary AR(1) processes
cites work: Maximum likelihood type estimation for nearly nonstationary autoregressive time series
cites work: The probabilities of large deviations for semimartingales
cites work: Best constants in martingale version of Rosenthal's inequality
cites work: Q3911791
cites work: Q5624460
cites work: Q3997990
cites work: Q4497390
cites work: Q4085018
cites work: Q3993391
cites work: On Large Deviation Asymptotics in Hilbert Space I
Recommended article: Q4505027
Recommended article: Q4289674
Recommended article: Q4512353
Recommended article: Limit theorems in the area of large deviations for some dependent random variables
Recommended article: Q3993391
Recommended article: Q4060904
Recommended article: Limit theorems for random excursions
Recommended article: Q5262331
description: scientific article; zbMATH DE number 1296187